Price of Bond given credit state matrix

"Consider a credit rating system consisting of four states, A,B,and D(default) with the following annual credit transition probability:

A= [ 0.7, 0.2, 0.1; 0.2, 0.5,0.3; 0,0,1]

For a company rated B, calculate the value of a 100 dollar bond with maturity three years from now and an annual coupon of $5."

Here's my thinking: Find A^2 and A^3, and the second row, third column is the probability of default for year 2, and 3 respectively.

So, Value = coupon payment*exp-(rate*time)*probability of default (for the three payments). The rate would come from second row, second column from matrix A. Would that be right?

Answer

Assuming recovery rate to be zero. The transition matrix is a table with three possible states for each group:

         A        B        D

    A   0.7      0.2      0.1

    B   0.2      0.5      0.3

    D   0.0      0.0      1.0

Since the question is about a company that is rated B now you expect that there is a 50% probability that the company will stay in B for the next period and that there is a 30% change it will default and that there is a 20% change that it will upgrade to A. Furthermore you assume the whole transition matrix is valid for two more periods. So to derive the calculations for this question I recommend you construct an event tree for a company that is B now using the multiplication rule for the probabilities:

                                              B
              0.2                            0.5                  0.3
               A                              B                    D
       0.14  0.04   0.02             0.1     0.25    0.15     0.0 0.0 0.3   
        A      B      D               A        B       D       A   B   D
      0.014  0.012  0.02            0.01     0.075   0.15             0.3
        D      D      D               D        D       D               D    

As you can see, there are three rows of number representing the probability of the rating for the three years you need to find the expected payments for, you can also call it the average cash flow, i.e. if you had e.g. 10000 of these bonds (each with a different company and their defaults are uncorrelated, a terrible representation of the real world non the less) you could expect these average payments multiplied with 10000. So the expected cash flow (average payment) of after the first year is: $c_1=0.7*\$5+0.3*\$0 = \$3.5$ i.e there is a 70% change you will receive \$5 and a 30% change you'll get nothing, on average you'll get \$3.5. For the second year you can see that 47% have defaulted on average up to that point so the cash flow becomes $c_2=(1-0.47)*\$5=\$2.65$. For the third year you receive the face value of the bond \$100 plus the coupon for the companies that haven't defaulted up to that point. (note that I don't show redundant calculations of the event tree for the 3rd year) so that the expected cash flow of the third year is: $c_3=(1-0.581)*(\$105) = \$43.995$.

Now you have obtained the time value of the expected payments and then you need to find the present value and set that as the price of the bond now. Assuming you have a yield curve (e.g. the funding rate), then you can obtain discount factors $D_t$ for the $t=1,2,$ and $3$ year points so that the bond price is: $$P_{t=0}=D_1*c_1+D_2*c_2+D_3*c_3$$ Assuming that you would recover something from the defaulted companies means you would need to add some amount to the bond price.

Back to your question on obtaining the default probabilities from matrix multiplication. The transition matrix is called A, not to be confused with the credit rating A. You can obtain the sum of the default probabilities for the second year by the matrix multiplication $AA$, it is located in row 2 column 3, i.e. the number 0.47 and in the same location is the number 0.581 for $(AA)A$ so you are absolutely right.